How do you find the linearization of f(x)=lnx f(x)=lnx at x=8?

1 Answer
Eddie
Feb 23, 2017

From simple geometry, plus some complicated things to do with general smoothness of the curve about x = 8x=8, we can say for "small" epsilonε that:

f(8 + epsilon) approx f(8) + epsilon f'(8)

And because f(x) = ln x then f'(x) = 1/x; and we can say that:

f(8 + epsilon) approx ln 8 + epsilon 1/8

We can test this in a calculator for f(8.1)

Actual Value: ln 8.1 = 2.0919

From Linearisation: ln 8 + 0.1 * 1/8 = 2.0919

!!

For f(8.9)

Actual Value: ln 8.9 = 2.186

From Linearisation: ln 8 + 0.9 * 1/8 = 2.192

:(

For f(20)

Actual Value: ln 20 approx 3

From Linearisation: ln 8 + 12 * 1/8 approx 3.6

:((

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